Figure 2: A Cooper pair formed of electrons with opposite momentum and spin.

The idea of a connection between mass generation and spontaneous symmetry breaking dates back to the work of Nambu (1960). This and other early work was largely based on analogies with the theory of superconductivity.

The essential physical basis of the BCS or Bardeen-Cooper-Schrieffer theory of superconductivity (Bardeen 1957), is the existence of an effective attractive force between electrons arising from their interaction with phonons in the lattice. This leads to the formation of bound pairs of electrons, Cooper pairs, in a spin-singlet state (see Figure 2). These Cooper pairs then undergo a Bose-Einstein condensation; in other words there is a macroscopic occupation of a single quantum state of a pair.

The Ginzburg-Landau model

A very convenient phenomenological model of a superconductor is provided by the Ginzburg-Landau model (Ginzburg 1950). This predates the BCS theory, but was later derived from it (in the London regime close to the critical temperature) by Gor'kov (1959).

The Ginzburg-Landau model involves a scalar order parameter field, \(\phi(t,\mathbf{x})\ .\) This field represents the wave function of the condensate of Cooper pairs; \(|\phi|^2\) is the number density of pairs in the condensate. Here \(\phi\) may be regarded as a composite of two electron fields, and so has charge \(2e\ .\) The Hamiltonian (in units with \(\hbar=1\)) is
\[\tag{1}
H=\int d^3\mathbf{x}
\left[\frac{1}{2m}\mathbf{D}\phi^*\cdot\mathbf{D}\phi+V(\phi)\right],\]

where
\[ \mathbf{D}\phi=\boldsymbol{\nabla}\phi-2ie\mathbf{A}\phi, \]
is the covariant derivative corresponding to the gauge transformations\[\tag{2}
\phi\to\phi e^{2ie\lambda},\qquad \mathbf{A}\to\mathbf{A}+\boldsymbol{\nabla}\lambda,\]

and the potential \(V(\phi)\) is a function only of \(\phi^*\phi\ .\) Near the critical temperature \(T_{\mathrm{c}}\) the field \(\phi\) is small, and \(V\) may be expanded in a power series, keeping only quadratic and quartic terms:
\[\tag{3}
V(\phi)=\alpha\phi^*\phi+\tfrac{1}{2}\beta(\phi^*\phi)^2.\]

The coefficients \(\alpha\) and \(\beta\) are temperature-dependent, and in particular \(\alpha\) changes sign at the critical temperature. For \(T<T_{\mathrm{c}}\ ,\) it is negative, so \(V\) has the 'sombrero' shape (see Figure 1) and its minima occur not at the symmetry point \(\phi=0\) but around the circle
\[\tag{4}
\phi^*\phi=\frac{-\alpha}{\beta}.\]

Consequently, the gauge symmetry (2) is spontaneously broken. The magnitude of the order parameter \(|\phi|\) is fixed by (4) but its phase is arbitrary. There is a family of degenerate ground states, labelled by this phase. Note that since the phase is canonically conjugate to the occupation number, a definite phase requires an indefinite number of pairs in the condensate.

Following the development of this theory, there was much discussion in the literature of the role of gauge invariance. The Ginzburg-Landau ground state does not respect this gauge symmetry, and nor does the ground state of the full Bardeen-Cooper-Schrieffer model. Today this is recognized as an example of the phenomenon of spontaneous symmetry breaking, but at the time it was seen by many as a defect of the theory.

Nambu (1960) gave a particularly clear account of the situation, using a modified form of the Hartree-Fock approximation. (See also Bogoliubov (1959).) He showed how the symmetry breaking is intimately connected with the energy gap. So long as the force between electrons is attractive, dissociating a Cooper pair requires an energy input. Thus the excitations of the system, quasiparticle states, are separated from the ground state by a finite energy gap; the energy \(E_{\mathbf{p}}\) of a quasiparticle of momentum \(\mathbf{p}\) tends to a non-zero limit as \(\mathbf{p}\to\mathbf{0}\ .\) Closely related is the fact that the quasiparticles do not have a definite charge (Bogoliubov (1959)): a quasiparticle is a linear combination of an electron of momentum \(\mathbf{p}\) and spin up, say, and a hole in the state of momentum \(-\mathbf{p}\) and spin down.

Analogous mechanism in particle physics

Nambu then went on to suggest that the masses of elementary particles might arise in a similar way; the vacuum state, like the superconducting ground state, might not respect the symmetries of the theory, and so elementary particles, like the quasiparticles in BCS theory, might acquire masses.

In particular, Nambu envisaged a theory of strong interactions involving a fundamental massless fermion field \(\psi(x)\) with a Lagrangian invariant under both ordinary and chiral global phase changes:
\[\tag{5}
\psi(x)\to e^{i\alpha}\psi(x),\qquad\text{and}\qquad
\psi(x)\to e^{\alpha\gamma_5}\psi(x)\]

(in a representation in which \(\gamma_5^2=-1\)). Correspondingly there would be both vector and axial vector conserved Noether currents,
\[\tag{6}
j^\mu=\bar\psi\gamma^\mu\psi,\qquad
j^\mu_5=\bar\psi i\gamma^\mu\gamma_5\psi.\]

He suggested that, like the superconducting energy gap, the nucleon mass arises from spontaneous breaking of the chiral symmetry.

Together, Nambu and Jona-Lasinio (1961) constructed a specific model with these characteristics, with four-fermion interactions, based on the Lagrangian density
\[\tag{7}
L=i\bar\psi\gamma^\mu\partial_\mu\psi
+g[(\bar\psi\psi)^2-(\bar\psi\gamma_5\psi)^2].\]

They then assumed that in the ground state or vacuum, the chiral symmetry is broken spontaneously by a non-zero expectation value \(\langle0|\bar\psi(x)\psi(x)|0\rangle=n\ ,\) say. Using techniques similar to those he employed to discuss superconductivity, they showed that this would indeed imply a nonzero mass for the 'quasiparticle', here identified as the nucleon.

Nambu-Goldstone bosons

Like the model of Goldstone (1961) (see main page), the
Nambu--Jona-Lasinio model also predicts the existence of a massless particle of spin zero, the Goldstone boson or Nambu-Goldstone boson, a nucleon-antinucleon bound state. This is a consequence of the broken chiral symmetry, and the particle in this case is a pseudoscalar. Since subjecting all the particles to a chiral rotation merely takes us from one degenerate vacuum state to another, and so requires no energy, it follows that applying a spatially varying chiral rotation with long wavelength requires very little energy; the energy goes to zero in the long-wavelength limit, which means the particle is massless.

It was clear that these were examples of a general phenomenon, described by what has come to be known as the Goldstone theorem. In the context of an explicitly relativistic theory, a general proof was provided by Goldstone, Salam & Weinberg (1962).

Nambu and Jona-Lasinio realized that as a consequence of the broken symmetry, their explanation for the nucleon mass would necessarily imply the existence of a massless particle of zero spin. Of course, no such particle was (or is) known, but they suggested that in fact the chiral symmetry is not quite exact but intrinsically weakly broken -- in addition to the larger spontaneous symmetry breaking effect. Then the would-be Nambu-Goldstone bosons would acquire small masses, and could be identified with the pions. (They pointed out that it would be easy to accommodate isospin within the model.) Although the Nambu--Jona-Lasinio model has been superseded, this identification is in fact in line with current theory. The nucleon is no longer regarded as a quasiparticle associated with a fundamental field; it is a composite of quarks. But it remains true that if the quarks were massless, then quantum chromodynamics would exhibit chiral symmetry, and the pions would be massless. The weak chiral symmetry breaking leads to a small pion mass.

Nambu and Jona-Lasinio also noted that in the case of a superconductor the particle-hole bound state that would play the role of the Nambu-Goldstone boson is not massless because of the existence of the long-range Coulomb forces, but rather becomes part of the plasmon oscillations.

The first non-Abelian gauge theory was proposed by Yang and Mills (1954) (and independently by Shaw (1955), though his work appears only in a Cambridge University PhD thesis). This was based on the idea of promoting the global SU(2) isospin symmetry to a local gauge symmetry, which requires the introduction of an isospin triplet of gauge fields. This was intended as a theory of strong interactions, and as such has of course been superseded by quantum chromodynamics. But it was soon realized that a similar idea might work for the weak interactions.

The original Fermi theory of weak interactions involved a direct four-fermion interaction (see Figure 3(a)). Following the proposal by Lee & Yang (1956) of parity non-conservation in weak interactions, confirmed the following year (Wu 1957), it was established that the weak interactions are of the \(V-A\) form, involving interactions between vector and axial-vector currents. This led Feynman & Gell-Mann (1958) and independently Sudarshan & Marshak (1958) to suggest that they might be mediated by charged intermediate vector bosons, \(W^\pm\) ( Figure 3(b)), raising the interesting possibility of a gauge theory of weak interactions. Indeed because of the similarity to electromagnetic interactions ( Figure 3(c)) it suggested the even more interesting possibility of a unified theory of weak and electromagnetic interactions, in which \((W^+,\gamma,W^-)\) would form a triplet of gauge bosons.

However, this hypothesis was faced with two immediate and severe obstacles. Firstly, to explain the short range and weakness at low energies of the weak interactions, the \(W^\pm\) had to have very large masses, whereas it was generally believed that gauge bosons were necessarily massless, like the photon.

The second problem arose from the parity violation in weak interactions. The \(W^\pm\) interacted not with a vector current but with a chiral current, a sum of terms of the form \(\bar\psi_1\gamma^\mu(1-i\gamma_5)\psi_2\ .\) So how could they be part of the same multiplet as the photon, with its parity-conserving interactions? The solution to this second problem was eventually found to require the enlargement of the gauge group from \(SU(2)\) to \(SU(2)\times U(1)\ ,\) with the introduction of another neutral gauge boson, \(Z^0\ ,\) as first suggested by Glashow (1961). This development, however, was peripheral to the subject of the present article, and will not be considered further here.

Both these problems clearly required some mechanism for breaking the symmetry between the \(W^\pm\) and the photon. It was natural to ask whether this, like superconductivity, could be an example of spontaneous symmetry breaking. But that idea raised a new problem, the possible appearance of massless scalar particles. The Goldstone theorem seemed to suggest that a spontaneously broken gauge theory would be plagued by two different kinds of massless particles, the gauge vector bosons themselves and the scalar Nambu-Goldstone bosons.

Masses of gauge bosons

In the early days it was commonly believed that one of the virtues of the gauge principle -- the idea of promoting global to local symmetries, with the introduction of gauge vector particles -- was the successful prediction of the vanishing photon mass.

Figure 4: The lowest-order photon self-energy diagram.

The first person to question this orthodoxy was Schwinger (1962). One form of the argument for a massless photon was that by virtue of gauge invariance, the self-energy of the photon (see Figure 4) would necessarily have the form
\[ \Pi_{\mu\nu}(p)=(g_{\mu\nu}p^2-p_\mu p_\nu)\Pi(p^2), \]
which seemed to suggest that it must vanish at \(p^2=0\ ,\) implying zero mass. Schwinger argued convincingly however that if the interactions were strong enough, then \(\Pi(p^2)\) might acquire a pole at \(p^2=0\ ,\) thus eliminating the prediction. A photon with stronger interactions could be massive.

Then Anderson (1963) pointed out that there are examples of this in condensed-matter physics. In an electron plasma, electromagnetic waves cannot propagate if their frequency is less than the plasma frequency \(\omega_{\mathrm{pl}}\ ,\) given by
\[\omega_{\mathrm{pl}}^2 = \frac{e^2 n_e}{\epsilon_0 m_e}, \]
where \(n_e\) is the electron number density and \(m_e\) the electron mass. In fact, the dispersion relation for this plasmon is for small \(\mathbf{k}\) of the form,
\[\omega^2 =\omega_{\mathrm{pl}}^2+u^2\mathbf{k}^2, \]
so in effect the photon has acquired a mass \(m_{\mathrm{pl}}=\hbar\omega_{\mathrm{pl}}/c^2\ .\) The same thing applies in a superconductor.

This work however was framed in language that was not then familiar to most particle physicists, and referred to models without relativistic invariance; the plasma or superconductor defines a preferred frame of reference. But, because of the work of Goldstone, Salam & Weinberg, it was widely believed among field theorists that in a fully relativistic theory the Goldstone theorem would unambiguously demand massless particles. (See, for example, Gilbert (1964).) Although Anderson had shown that in the nonrelativistic context the massless gauge bosons and Nambu-Goldstone bosons could combine to form a massive vector particle (a result already foreshadowed by Nambu and Jona-Lasinio (1961)), most particle theorists believed that could not happen in a relativistic theory.

The final step of showing that indeed the mechanism could work in a relativistic theory was taken independently by three groups: Englert & Brout, Higgs, and Guralnik, Hagen & Kibble, approaching the problem from three rather different perspectives. The simplest and most direct argument was that of Higgs (1964b), where he exhibited a very simple U(1) model, now often called the Abelian Higgs model (see the article Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism). Prior to that (Higgs 1964a) he had made the important observation that in a gauge theory formulated in the radiation gauge, the choice of gauge removes the explicit relativistic invariance, and thus renders the Goldstone theorem inapplicable. Higgs's treatment of the model was purely classical, but essentially the same model was considered in a quantum mechanical way by both the other groups. Englert & Brout (1964), whose paper was the first to be published, based their argument on a calculation of the vacuum polarization in lowest-order perturbation theory about the assumed symmetry-breaking vacuum state. They also pointed out that the same mechanism could operate in more general non-Abelian models, and gave simple examples. Guralnik, Hagen & Kibble (1964), on the other hand, used an operator formalism, concentrating on the role of the conservation law and the precise way in which the Goldstone theorem can be evaded. (For more detail of the history leading up to this paper, see Guralnik 2011.) However, the conclusions of all three were essentially the same.

Later developments

There were several significant developments in the years immediately after 1964. Higgs (1966) studied his model quantum mechanically in some detail, evaluating transition and decay amplitudes for the model in lowest-order perturbation theory, and in particular discussing the induced symmetry-breaking effects that would occur if the scalar field were coupled to other fields. Streater (1965) discussed the mechanism from the point of view of axiomatic field theory. The details of the application to a non-Abelian gauge theory were studied by Kibble (1967), showing how the numbers of massive and massless states were related to the symmetry-breaking pattern. For a comprehensive review of symmetry breaking in both field theory and condensed matter, see Guralnik et al (1967).

Initially, the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism was envisaged as having as much, or more, relevance to the development of a gauge theory of strong as of weak interactions. But of course its most significant application was to the development of the unified electroweak theory by Weinberg (1967) and Salam (1968), incorporating the model developed earlier by Glashow (1961).

Both Weinberg and Salam had conjectured that the theory was renormalizable, but with no proof. The final, very important element was the proof of renormalizability of spontaneously broken gauge theories by 't Hooft (1971).

Historically, the primary focus of the mechanism was on how the gauge bosons acquire non-zero mass. The fact that there are also massive Higgs bosons was incidental. However, their existence has now become of great significance, because the Higgs boson is at present (2009) the only component of the standard model whose existence has not been experimentally confirmed1. Finding them (or some alternative) will be a key advance.

1 On 4 July 2012 a previously unknown boson of mass $\sim$ 125 GeV was confirmed to exist by the ATLAS and CMS teams at the Large Hadron Collider at CERN. This particle has been tentatively confirmed as a Higgs boson by CERN on 14 March 2013, although it remains an open question whether this is the Higgs boson of the Standard Model of particle physics, or possibly the lightest of several bosons predicted in some theories that go beyond the Standard Model.

Further reading

Weinberg, S (1996). The quantum theory of fields, vol. II: modern applications. Cambridge University Press, Cambridge. Chapter 21. ISBN 0-521-55001-7. This book contains a particularly interesting account of superconductivity from the standpoint of symmetry breaking, showing that many of the key features of superconductors can be derived just from the assumption of spontaneous symmetry breaking.

Zee, A (2003). Quantum field theory in a nutshell. Princeton University Press, Princeton, NJ. ISBN 0-691-01019-6.